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God created the real numbers

ethanheilman.com

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Re: God created the real numbers

#11

Post by nh23423fefe »

andrewla wrote: post #5
empath75 wrote: post #4

Earlier quoted context omitted.

> But the Cantor vision of the real numbers is just wrong and completely unphysical. They're unphysical, and yet the very physical human mind can work with them just fine. They're a perfectly logical construction from perfectly reasonable axioms. There are lots of objects in math which aren't physically realizable. Plato would have said that those sorts of objects are more real than anything which actually exists in…

There are two things being talked about here, and worth teasing them out. On the one hand, this article is talking about the hierarchy of "physicality" of various mathematical concepts, and they put Cantor's real numbers at the floor. I disagree with that specifically; two quantities are interestingly "unequal" only at the precision where an underlying process can distinguish them. Turing tells us that any underlying…

> the idea that there are the same number of integers as even integers is a stupid one that in the end does not lead anywhere useful

it leads to the idea that measuring 2 sets via a bijection is a better idea than measuring via containment

Re: God created the real numbers

#12

Post by tialaramex »

empath75 wrote: post #4
andrewla wrote: post #3

I'm an enthusiastic Cantor skeptic, I lean very heavily constructivist to the point of almost being a finitist, but nonetheless I think the thesis of this article is basically correct. Nature and the universe is all about continuous quantities; integral quantities and whole numbers represent an abstraction. At a micro level this is less true -- elementary particles specifically are a (mostly) discrete phenomenon, but…

> But the Cantor vision of the real numbers is just wrong and completely unphysical. They're unphysical, and yet the very physical human mind can work with them just fine. They're a perfectly logical construction from perfectly reasonable axioms. There are lots of objects in math which aren't physically realizable. Plato would have said that those sorts of objects are more real than anything which actually exists in…

> They're unphysical, and yet the very physical human mind can work with them just fine

Nah, you're likely thinking of the rationals, which are basically just two integers in a halloween costume. Ooh a third, big deal. The overwhelming majority of the reals are completely batshit and you're not working with them "just fine" except in some very hand wavy sense.

Re: God created the real numbers

#13

Post by athrowaway3z »

Can't say that I'm completely in the headspace to follow the argument, but wanted to add my 2 cents from a few years ago.

Integers come into existence long before god - as the only presumption required is a difference between one thing and another (or nothing). The integers also create infinite gaps. The primes.

So no - I do not think reals are closer to the divine. They require we import infinity twice to be defined, and I'm undecided on whether our reality has unbounded 'precision' like that - or if 'just' an infinite number of discrete units.

Re: God created the real numbers

#14

Post by chasd00 »

andrewla wrote: post #3

I'm an enthusiastic Cantor skeptic, I lean very heavily constructivist to the point of almost being a finitist, but nonetheless I think the thesis of this article is basically correct. Nature and the universe is all about continuous quantities; integral quantities and whole numbers represent an abstraction. At a micro level this is less true -- elementary particles specifically are a (mostly) discrete phenomenon, but…

> The idea of arbitrary precision is intrinsically broken in physical reality.

you said a lot and i probably don't understand but doesn't pi contradict this? pi definitely exists in physical reality, wherever there is a circle, and seems to be have a never ending supply of decimal points.

Re: God created the real numbers

#15

Post by LegionMammal978 »

chasd00 wrote: post #14
andrewla wrote: post #3

I'm an enthusiastic Cantor skeptic, I lean very heavily constructivist to the point of almost being a finitist, but nonetheless I think the thesis of this article is basically correct. Nature and the universe is all about continuous quantities; integral quantities and whole numbers represent an abstraction. At a micro level this is less true -- elementary particles specifically are a (mostly) discrete phenomenon, but…

> The idea of arbitrary precision is intrinsically broken in physical reality. you said a lot and i probably don't understand but doesn't pi contradict this? pi definitely exists in physical reality, wherever there is a circle, and seems to be have a never ending supply of decimal points.

> wherever there is a circle,

Is there a circle in physical reality? Or only approximate circles, or things we model as circles?

In any case, a believer in computation as reality would say that any digit of π has the potential to exist, as the result of a definite computation, but that the entirety does not actually exist apart from the process used to compute it.

Re: God created the real numbers

#16

Post by alphazard »

andrewla wrote: post #3

I'm an enthusiastic Cantor skeptic, I lean very heavily constructivist to the point of almost being a finitist, but nonetheless I think the thesis of this article is basically correct. Nature and the universe is all about continuous quantities; integral quantities and whole numbers represent an abstraction. At a micro level this is less true -- elementary particles specifically are a (mostly) discrete phenomenon, but…

> Nature and the universe is all about continuous quantities

One could argue that nature always deals in discrete quantities and we have models that accurately predict these quantities. Then we use math that humans clearly created (limits) to produce similar models, except they imagine continuous inputs.

Re: God created the real numbers

#17

Post by andrewla »

blueplanet200 wrote: post #7
andrewla wrote: post #3

I'm an enthusiastic Cantor skeptic, I lean very heavily constructivist to the point of almost being a finitist, but nonetheless I think the thesis of this article is basically correct. Nature and the universe is all about continuous quantities; integral quantities and whole numbers represent an abstraction. At a micro level this is less true -- elementary particles specifically are a (mostly) discrete phenomenon, but…

>I'm an enthusiastic Cantor skeptic A skeptic in what way? He said a lot.

Here I'm referring to the cloud of things that Hilbert called "Cantor's Paradise". Basically everything around the notion of cardinality of infinities.

Re: God created the real numbers

#18

Post by LegionMammal978 »

Personally, I like to split the difference: the physical continuum definitely exists, to whatever extent any physical thing exists, but the real number line (and indeed the completed inductive set of integers) may just be a human-constructed fiction. The physical continuum is not necessarily identical to the real continuum; the latter is just a very useful model that lets us do human things like calculus.

(And the discrepancy might not be in the physical continuum being simpler than the mathematical reals, as some here postulate, but rather in the continuum being far stranger than the reals, in ways we may never observe nor comprehend.)

Re: God created the real numbers

#19

Post by NoahZuniga »

andrewla wrote: post #3

I'm an enthusiastic Cantor skeptic, I lean very heavily constructivist to the point of almost being a finitist, but nonetheless I think the thesis of this article is basically correct. Nature and the universe is all about continuous quantities; integral quantities and whole numbers represent an abstraction. At a micro level this is less true -- elementary particles specifically are a (mostly) discrete phenomenon, but…

You know it wouldn't be possible for us to tell the difference between a rational universe (one where all quantities are rational numbers) and a real universe (one where you can have irrational quantities).

The standard construction for the real numbers is to start with the rationals and "fill in all the holes". So why even bother with filling in the holes and instead just declare God created the rationals?

Re: God created the real numbers

#20

Post by andrewla »

nh23423fefe wrote: post #11
andrewla wrote: post #5

Earlier quoted context omitted.

There are two things being talked about here, and worth teasing them out. On the one hand, this article is talking about the hierarchy of "physicality" of various mathematical concepts, and they put Cantor's real numbers at the floor. I disagree with that specifically; two quantities are interestingly "unequal" only at the precision where an underlying process can distinguish them. Turing tells us that any underlying…

> the idea that there are the same number of integers as even integers is a stupid one that in the end does not lead anywhere useful it leads to the idea that measuring 2 sets via a bijection is a better idea than measuring via containment

That a bijection exists is incredibly useful. But the idea of "measuring" infinite sets in the cardinality sense is not very interesting or useful.
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